The document discusses Church encoding (B) and defines Bn as repeated function application. It shows Bn exhibits periodicity, with B(k) = B(k-4) for k ≥ 10. It also explores properties of functions under B, including proving Bn exhibits more complex periodic patterns for larger n.

1. 8 : L OFOF :FD O
0LL FA?OF C B O MIN
! 4 !
( l
995 (& , ( ! - 12 (&
b brb
S SSSe
S c c
b bmb brb
SSSSSSS v
e
(.)(.)(.)(.)(.)(.)(.)(.)(.)(.) == (.)(.)(.)(.)(.)(.)

2. s B v
t m
B f g = f ◦ g
v B ≡ λ f g x. f (g x)

3. n
λ f g x1 ... xn. f (g x1 ... xn)
g n l
Bn
o s
B2 = B◦B = λ f g x1 x2. f (g x1 x2)
B3 = B◦B◦B = λ f g x1 x2 x3. f (g x1 x2 x3)
B = (◦) Bn
B r
B2 = B◦B = B B B
B3 = B◦B◦B = B (B B B) B
= B2 B B B B B
= B B B B B B B B
B
n
c v w
f

4. Bn
u o m
B u B(k)
B(1) = B B(k+1) = B(k) B
Bn
B(k) o m
B2 = B B B = B(3) B3 = B B B B B B B B = B(8)
B4 B(k) o s
B(k) l
(
g
c u
B(1)=λxyz.x(yz)
B(2)=λxyzw.xy(zw)
B(3)=λxyzw.x(yzw)
B(4)=λxyzwv.xyz(wv)
B(5)=λxyzw.x(y(zw))
B(6)=λxyzwv.x(yz)(wv)
B(7)=λxyzwv.xy(zwv)
B(8)=λxyzwv.x(yzwv)
B(9)=λxyzwvu.xyzw(vu)
B(10)=λxyzwv.x(yz)(wv)
B(11)=λxyzwv.xy(zwv)
：
B4 B(1) B(9) u
k ≥ 10 B(k) = B(k−4)
h B4 B(k) o
a (.)(.)(.)(.)(.)(.)(.)(.)(.)(.)==(.)(.)(.)(.)(.)(.)

5. Bn
u o m
B u B(k)
B(1) = B B(k+1) = B(k) B
Bn
B(k) o m
B2 = B B B = B(3) B3 = B B B B B B B B = B(8)
B4 B(k) o s
B(k) l
(
g
c u
B(1)=λxyz.x(yz)
B(2)=λxyzw.xy(zw)
B(3)=λxyzw.x(yzw)
B(4)=λxyzwv.xyz(wv)
B(5)=λxyzw.x(y(zw))
B(6)=λxyzwv.x(yz)(wv)
B(7)=λxyzwv.xy(zwv)
B(8)=λxyzwv.x(yzwv)
B(9)=λxyzwvu.xyzw(vu)
B(10)=λxyzwv.x(yz)(wv)
B(11)=λxyzwv.xy(zwv)
：
B4 B(1) B(9) u
k ≥ 10 B(k) = B(k−4)
h B4 B(k) o
a (.)(.)(.)(.)(.)(.)(.)(.)(.)(.)==(.)(.)(.)(.)(.)(.)

6. X ρ
X ρ i j u X(i) = X(j)
ρC l
ρ
B : B(10) = B(6)
K (= λxy.x) : K(3) = K(1)
C (= λxyz.xzy) : C(4) = C(3)
B m B v [
B B ρ l (B B)(52) = (B B)(32)
u d
B B B ρ l i≠j (B B)(i)≠(B B)(j)
y )
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
F
B(9) B(8)K(1) K(2)
→C(1)→C(2)→C(3)
↻

7. Bn
B ρ
ρ B B B Bn
B
n ≥ 2 [
n = 0 : B(10) = B(6)
n = 1 : (B B)(52) = (B B)(32)
n = 2 : (B2 B)(294) = (B2 B)(258)
n = 3 : (B3 B)(10036) = (B3 B)(4240)
n = 4 : (B4 B)(622659) = (B4 B)(191206)
n = 5 : (B5 B)(1000685878) = (B5 B)(766241307)
n = 6 : (B6 B)(2980054085040) = (B6 B)(2641033883877)
uT u n=4 um o n ]
n=6 (& , mr
u ) n k c R

8. B ρ v
ρ ρ
B
B B
B2 B
B3 B
B4 B
B5 B
B6 B
B B B

9. B ρ v
ρ ρ
B
B B
B2 B
B3 B
B4 B
B5 B
B6 B
Bn
B.
B B B
& B X n
X n ρ i X n Bn
B

10. h
C n = 6
o l
B l
C B ρ l
B λ l
l
& B X n
X n ρ i X n Bn
B

11. X n B o
X n ρ X =βη Bn
B
f

12. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

13. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

14. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

15. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

16. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

17. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

18. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

19. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)

20. ktp m
X(i) = X(2i) i l
X(m)=X(m+k) (k>0) X(i) = X(2i) i C
B u n
B {
X(n) X(n+1) l
i = mk
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
B(8)B(9)02361 o d
l d {

21. B
λ l β
B }
B (B B B) B (B B) (B B)
B B (B B) B (B (B B)) B
B
B l } r
5446 , 9
4 PA F B w
(Bp1
B)◦(Bp2
B)◦…◦(Bpk
B)
uTp1 ≥ p2 ≥ … ≥ pk v
≈ [ p1, p2, ... , pk ]
[ 1, 1 ]
[ 2, 0 ]

22. X n B o
X n ρ h X =βη Bn
B
g

23. ρ v
B v λ v
λx1x2...xN. x1 x2 ... xN l d 8
B 34 I
B X u TX r
TX ] v
:X ∈ TX
:e ∈ TX ⇒ e X ∈ TX
:e ∈ TX ⇒ spine(e) < size(X)
X(k) ∈ TX
size(X(k)) ≤ size(X(k+1))
size(e) spine(e)
B = λxyz.x(yz) B B = λxyzw.xy(zw) B (B B) B = λxyzw.x(y(zw))

24. B X ρ v
X = (Bk B)(k+2)n (k≥0, n≥1)
TX
(⏟
t
t[
k+1
[
⏟
t
t
k+1
n
[
t :=
t
t
t
[
⏟
k+1
TX :=
⏟
[
k+1
[
⏟
k+1
n
[
[
⏟
k
X =
:X ∈ TX
:e ∈ TX ⇒ e X ∈ TX
:e ∈ TX ⇒ spine(e) < size(X)

25. B ρ v
s m
)
https://github.com/ksk/Rho
ρ ρ
B ≈ [0]
B B ≈ [1]
B2 B ≈ [2]
B3 B ≈ [3]
B4 B ≈ [4]
B5 B ≈ [5]
B6 B ≈ [6]
(B2B)2◦(B B)2◦B2
≈ [2,2,1,1,0,0]
(Bk B)(k+2)n ≈ [k,…,k]
⏟(k+2)n
(B B)3◦B3
≈ [1,1,1,0,0,0]c R